The invention relates generally to the field of mathematical modeling of random or stochastic systems, and more particularly to modeling of the behavior of financial markets and instruments.
Over the past thirty years many risky asset models have been proposed, but these are generally deficient in that they do not adequately explain key features from the statistical evidence related to the issues of heavy tails and long-range dependence (LRD) in financial time series. With regard to long-range dependence, there have been problems even with understanding the nature of the characteristic, let alone its assessment.
In particular, the Black-Scholes model for the price of a risky asset, which is widely used in the finance industry, makes two fundamental assumptions which are seriously flawed. They are:
(A) the differences of log(price) over time are xe2x80x9cnormallyxe2x80x9d distributed (i.e., the differences are a Gaussian process); and
(B) the differences of log(price) over different time periods are independent.
In order to elaborate on the flaws of the conventional model, it is helpful to introduce some notation. First, a value Xt is defined as log Ptxe2x88x92log Ptxe2x88x921, where Pt is the asset price at time t. Then, according to assumption (A), the values {Xt, t=1,2, . . . } are normally distributed and, according to assumption (B), the values are independent.
The problem with assumption (A) can readily be seen by studying real data. For normally distributed data, values which are more than 5 standard deviations away from the mean are very rare, with a probability less than 10xe2x88x926. However, empirical studies of actual risky asset data demonstrate that observations this far away from the mean are not uncommon. The data exhibit a pronounced leptokurtic distribution (much higher peaks and heavier tails than Gaussian). In fact, large values can often be observed out to 10 standard deviations. To put these observations into perspective, consider that a 10xe2x88x926 probability corresponds to a once-in-4000-years event for a normal (i.e., Gaussian) distribution of transaction data collected 5 days per week.
The problem with assumption (B) can readily be seen by examining lagged autocorrelations of the {Xt} process. While estimated values of corr(Xt, Xt+k) die out quickly as k increases, the values of corr(|Xt|,|Xt+k|) and corr(Xt2, Xt+k2) remain significantly different from zero for long periods. In fact, non-zero values can be observed over periods of many years for some series. This has been convincingly illustrated by analysis of daily returns data from the SandP 500 index, the Nikkei index, foreign exchange rates (DM/US$), and Chevron Stock. See, e.g., Z. Ding and C. W. J. Granger, Modeling volatility persistence of speculative returns: A new approach, 73 J. Enconometrics 185 (1996); C. W. J. Granger and Z. Ding, Varieties of long memory models, 73 J. Econometrics 61 (1996). In the case of the absolute values from the SandP series, for example, the lagged autocorrelations are significantly different from zero for lags up to 2700, which is more than 10 years. The absolute values and squares of the returns exhibit LRD, classically defined (for stationary finite variance processes) as existing if xcexa3k=0∞ xcex3k diverges, xcex3k being the autocorrelation at lag k. For a discussion of the definition and its extensions see C. C. Heyde and Y. Yang, On defining long-range dependence, 34 J. Applied Prob. 939 (1997). For a comprehensive overview of the subject of LRD see J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, New York (1994).
Yet, according to assumption (B), the functions corr(Xt, Xt+k), corr(|Xt|,|Xt+k|), and corr(Xt2, Xt+k2) should all be zero for all kxe2x89xa71. In other words, assumption (B) is belied by the fact that the corr(|Xt|,|Xt+k|) and corr(Xt2, Xt+k2) functions exhibit strong dependence effects when applied to real, observed data. In addition, the actual data form a time series with high volatility and xe2x80x9cintermittencyxe2x80x9d (clustering of extremes) quite unlike white noise. Failure to take these effects into account can adversely affect almost all probabilistic assessments involving price movements.
It is now widely appreciated that heavier tails than the normal are a necessary model feature for marginal distributions of returns. However, models with finite moments of all orders are still commonly advocated. Detailed comparisons using market index data have been given in S. R. Hurst, E. Platen, and S. R. Rachev, Subordinated Markov models: a comparison, Fin. Eng. Japanese Markets 97 (1997); S. R. Hurst and Platen, E., The marginal distribution of returns and volatility, in L1xe2x80x94Statistical Procedures and Related Topics 301 (IMS Lecture Notes-Monograph Series, Vol. 31, IMS, Hayward, Calif., Y. Dodge ed., 1997). They favor a t-distribution with degrees of freedom v typically in the range 3-5. This, of course, implies an infinite k-th moment for kxe2x89xa7v.
The above-described feature of infinite k-th moment has been observed in actual financial data. For example, multiple tests on daily data from the well-known SandP 500 Index lead to the conclusion that the 4th moment is infinite. See, e.g., T. Mikosch, Heavy tails in finance, 57 Bull. Internat. Statist. Inst. 109 (Book 2) (1997); Y. Yang, Long Range Dependence in the Study of Time Series with Finite or Infinite Variances, Ph.D. Dissertation, Columbia University (1998).
Most approaches to the marginal distribution of returns have involved the use of parametric families and the choice of a best fit within a family. Few authors have investigated general questions, such as whether particular moments are finite. Part of the difficulty has been a general supposition that tail behavior would be a power function, at least asymptotically, and that estimating the power index is problematic. Indeed, the usual method, based on the Hill estimator, is founded on order statistic theory for independent and identically distributed (hereinafter xe2x80x9ciidxe2x80x9d) samples and can suffer from major shortcomings outside that context. See for example, the xe2x80x9cHill Horror Plotxe2x80x9d and the attendant discussion in S. Resnick, Heavy tail modeling and teletraffic data, 25 Ann. Statist. 1805 (1997).
Accordingly, it is an object of the invention to provide a new mathematical model with better capability of predicting the behavior of random or stochastic processes such as the behavior of financial markets and instruments. The new model replaces the time parameter of the Black-Scholes model with a fractal activity time, which is a random activity time process having a heavy-tailed distribution, LRD, and stationary differences. The fractal activity time is standardized so that its mean value corresponds to the time parameter used in the Black-Scholes model.
In accordance with the invention, it is possible to test for finiteness of moments based on asymptotics obtained via the use of the ergodic theorem if the assumption of stationarity of the returns is retained. In particular, if xcexa3i=1n|Xi|p/n converges as nxe2x86x92∞, then E|X|p less than ∞, where E|X|p denotes the mean of |X|p. If, on the other hand, max1xe2x89xa6ixe2x89xa6n|Xi|p/xcexa3i=1n|Xi|p stays away from zero as nxe2x86x92∞, then E|X|p=∞. Plots as n increases for large data sets should give a clear picture.
In a method according to the invention, a random process is modeled by providing time-parameter data which corresponds to a fractal activity time process or an approximation thereof. The time-parameter data is used to determine whether the random process exhibits fractal scaling or self-similarity, in which case it is determined that the random process can be validly described using a model based on the time-parameter data.
Further objects, features, and advantages of the invention will become apparent from the following detailed description taken in conjunction with the accompanying figures showing illustrative embodiments of the invention.